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algebraic coding theory IN THE UNDERGRADUATE CURRICULUM

AMERICAN MATHEMATICAL MONTHLY 1984年第8期91卷 509-513页

作者： BRINN, LW

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关键词： algebraic coding theory Morse code Linear algebra Discrete mathematics Error correcting codes Mathematical induction College mathematics Polynomials Computer science

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Quantum message authentication based on algebraic coding theory

Quantum message authentication based on algebraic coding the...

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Conference on Quantum Optics and Applications in Computing and Communications II

作者： Yang, L Hu, L Feng, DG Chinese Acad Sci Grad Sch State Key Lab Informat Secur Beijing 100039 Peoples R China

ISBN: (纸本)0819455865

We present a new type of authentication scheme for quantum message based on algebraic coding theory and quantum computation operations between different quantum registers. The results are that if the pre-coding generator matrix in SN-S code is public, the quantum scheme is a public-key data integrity scheme;if it is secret, the quantum scheme is a hybrid data origin authentication scheme. The advantage of this scheme is that the public and secret keys are merely some classical data.

关键词： quantum authentication quantum computation algebraic coding theory public-key cryptosystem

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New constructions of abelian non-cyclic orbit codes based on parabolic subgroups and tensor products

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FINITE FIELDS AND THEIR APPLICATIONS 2025年 103卷

作者： Askary, Soleyman Biranvand, Nader Shirjian, Farrokh Imam Ali Univ Fac Sci Tehran Iran

Orbit codes, as special constant dimension subspace codes, have attracted much attention due to their applications for error correction in random network coding. They arise as orbits of a subspace of Fqn under the action of some subgroup of the finite general linear group GLn(q). The main contribution of this paper is to propose new methods for constructing large non-cyclic orbit codes. First, using the subgroup structure of maximal subgroups of GLn(q), we propose a new construction of an abelian non-cyclic orbit codes of size qk with k <= n/2. The proposed code is shown to be a partial spread which in many cases is close to the known maximum-size codes. Next, considering a larger framework, we introduce the notion of tensor product operation for subspace codes and explicitly determine the parameters of such product codes. The parameters of the constructions presented in this paper improve the constructions already obtained in [6] and [7]. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

关键词： Random linear network coding algebraic coding theory Abelian orbit code Subspace code Linear groups

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THE ALGEBRA OF THE FINITE FOURIER-TRANSFORM AND coding theory

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TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY 1985年第1期287卷 253-273页

作者： TOLIMIERI, R Department of Electrical Engineering City College of New York CUNY New York NY 10031 United States

Abstract: The role of the finite Fourier transform in the theory of error correcting codes has been explored in a recent text by Richard Blahut. In this work we study how the finite Fourier transform relates to certain polynomial identities involving weight enumerator polynomials of linear codes. These include the generalized MacWilliams identities and theorems originally due to ${\text {R}}$. Gleason concerning polynomial algebras containing weight enumerator polynomials. The Heisenberg group model of the finite Fourier transform provides certain algebras of classical theta functions which will be applied to reprove Gleason’s results.

关键词： Mathematical theorems Polynomials Fourier transformations Homomorphisms Subalgebras algebraic coding theory Vector spaces

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Fuzzy Logics and Fuzzy Model theory

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ALGEBRA AND LOGIC 2015年第1期54卷 74-80页

作者： Pal'chunov, D. E. Yakhyaeva, G. E. Sobolev Inst Math Novosibirsk 630090 Russia Novosibirsk State Univ Novosibirsk 630090 Russia

The article presents an overview on the modern research of fuzzy logics and fuzzy model theory in the algebraic systems. It examines the concept of fuzzy in the proposed approximate reasoning of the valued logics in t... 详细信息

关键词： FUZZY logic MATHEMATICAL models algebraic coding theory REASONING (Logic) PROBABILITY theory

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Quantum theory as a Critical Regime of Language Dynamics

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FOUNDATIONS OF PHYSICS 2015年第10期45卷 1341-1350页

作者： Grinbaum, Alexei CEA Saclay IRFU LARSIM F-91191 Gif Sur Yvette France

Some mathematical theories in physics justify their explanatory superiority over earlier formalisms by the clarity of their postulates. In particular, axiomatic reconstructions drive home the importance of the composition rule and the continuity assumption as two pillars of quantum theory. Our approach sits on these pillars and combines new mathematics with a testable prediction. If the observer is defined by a limit on string complexity, information dynamics leads to an emergent continuous model in the critical regime. Restricting it to a family of binary codes describing 'bipartite systems,' we find strong evidence of an upper bound on bipartite correlations equal to 2.82537. This is measurably different from the Tsirelson bound. The Hilbert space formalism emerges from this mathematical investigation as an effective description of a fundamental discrete theory in the critical regime.

关键词： Reconstruction of quantum theory algebraic coding theory Critical phenomena Continuity Tsirelson bound Bipartite correlations

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ERROR-CORRECTING CODES AND INVARIANT theory - NEW APPLICATIONS OF A 19TH-CENTURY TECHNIQUE

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AMERICAN MATHEMATICAL MONTHLY 1977年第2期84卷 82-107页

作者： SLOANE, NJA BELL TEL LABS INC MURRAY HILLNJ 07974

An unfashionable nineteenth century technique, invariant theory, has recently been used to study error-correcting codes. This technique is potentially of much wider application, is very powerful, often produces startl... 详细信息

关键词： Mathematical theorems Binary codes Combinatorics algebraic coding theory Perfect codes Degrees of polynomials Mathematical lattices

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Low-rank parity-check codes over Galois rings

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DESIGNS CODES AND CRYPTOGRAPHY 2021年第2期89卷 351-386页

作者： Renner, Julian Neri, Alessandro Puchinger, Sven Tech Univ Munich TUM Inst Commun Engn Munich Germany Tech Univ Denmark DTU Dept Appl Math & Comp Sci Lyngby Denmark

Low-rank parity-check (LRPC) codes are rank-metric codes over finite fields, which have been proposed by Gaborit et al. (Proceedings of the workshop on coding and cryptography WCC, vol 2013, 2013) for cryptographic applications. Inspired by a recent adaption of Gabidulin codes to certain finite rings by Kamche et al. (IEEE Trans Inf theory 65(12):7718-7735, 2019), we define and study LRPC codes over Galois rings-a wide class of finite commutative rings. We give a decoding algorithm similar to Gaborit et al.'s decoder, based on simple linear-algebraic operations. We derive an upper bound on the failure probability of the decoder, which is significantly more involved than in the case of finite fields. The bound depends only on the rank of an error, i.e., is independent of its free rank. Further, we analyze the complexity of the decoder. We obtain that there is a class of LRPC codes over a Galois ring that can decode roughly the same number of errors as a Gabidulin code with the same code parameters, but faster than the currently best decoder for Gabidulin codes. However, the price that one needs to pay is a small failure probability, which we can bound from above.

关键词： Galois rings Low-rank parity-check codes Rank-metric codes algebraic coding theory

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A new scheme for covert communication via 3G encoded speech

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COMPUTERS & ELECTRICAL ENGINEERING 2012年第6期38卷 1490-1501页

作者： Miao, Haibo Huang, Liusheng Chen, Zhili Yang, Wei Al-hawbani, Ammar Univ Sci & Technol China Dept Comp Sci & Technol Hefei 230026 Peoples R China Univ Sci & Technol China Suzhou Inst Adv Study Suzhou 215123 Peoples R China

Mobile communication through 3G network has grown rapidly in recent years. It might be of interest to transmit secret messages over 3G voice channels. In this paper, we introduce a new covert communication scheme via Adaptive Multi-Rate Wideband (AMR-WB) encoded speech. An adaptive suboptimal pulse combination constrained (ASOPCC) method is presented to embed data on compressed speech signal of AMR-WB codec. The method takes advantage of the "redundancy", created by non-exhaustive search of algebraic code-book, to encode secret information. An embedding factor eta is used to control embedding bits. By properly setting eta, ASOPCC can offer a better trade-off between speech quality and embedding capacity in the process of coding mode switching. Experimental results show that the proposed method is quite promising for both high capacity and good imperceptivity. Although ASOPCC is only applied to AMR-WB codec in this article, it can be further used by any other speech coding based on algebraic Coded Exited Linear Prediction (ACELP). (C) 2012 Elsevier Ltd. All rights reserved.

关键词： SPEECH coding MOBILE communication systems COMPUTER networks algebraic coding theory LINEAR systems PREDICTION models CONSTRAINT satisfaction (Artificial intelligence)

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Efficient multivariate low-degree tests via interactive oracle proofs of proximity for polynomial codes

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DESIGNS CODES AND CRYPTOGRAPHY 2023年第3期91卷 1111-1151页

作者： Augot, Daniel Bordage, Sarah Nardi, Jade Ecole Polytech CNRS UMR 7161 Inst Polytech Paris LIX Palaiseau France Inria Palaiseau France Univ Rennes CNRS IRMAR UMR 6625 Vannes France

We consider the proximity testing problem for error-correcting codes which consist in evaluations of multivariate polynomials either of bounded individual degree or bounded total degree. Namely, given an oracle function f : L-m -> F-q, where L subset of F-q, a verifier distinguishes whether f is the evaluation of a low-degree polynomial or is far (in relative Hamming distance) from being one, by making only a few queries to f. This topic has been studied in the context of locally testable codes, interactive proofs, probalistically checkable proofs, and interactive oracle proofs. We present the first interactive oracle proofs of proximity (IOPP) for tensor products of Reed-Solomon codes (evaluation of polynomials with bounds on individual degrees) and for Reed-Muller codes (evaluation of polynomials with a bound on the total degree) that simultaneously achieve logarithmic query complexity, logarithmic verification time, linear oracle proof length and linear prover running time. Such low-degree polynomials play a central role in constructions of probabilistic proof systems and succinct non-interactive arguments of knowledge with zero-knowledge. For these applications, highly-efficient multivariate low-degree tests are desired, but prior probabilistic proofs of proximity required super-linear proving time. In contrast, for multivariate codes of length N, our constructions admit a prover running in time linear in N and a verifier which is logarithmic in N. Our constructions are directly inspired by the IOPP for Reed-Solomon codes of [Ben-Sasson et al., ICALP 2018] named "FRI protocol". Compared to the FRI protocol, our IOPP for tensor products of Reed-Solomon codes achieves the same efficiency parameters. As for Reed-Muller codes, for fixed constant number of variables m, the concrete efficiency of our IOPP for Reed-Muller codes compares well, all things equal.

关键词： algebraic coding theory Reed-Solomon codes Product codes Reed-Muller codes Low degree testing Interactive proof systems

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